Associate Professor Dr. Raed Ibraheem Hamed
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1 Associate Professor Dr. Raed Ibraheem Hamed University of Human Development, College of Science and Technology Computer Science Department Department of Computer Science _ UHD 1
2 What this Lecture is about: Traversals Traversing Trees Types of traversals Search Trees (BST) How to search a binary tree? Some terminology of Binary Trees Department of Computer Science _ UHD 2
3 Binary Tree Traversal Methods It's unclear how we should print a tree. Top to bottom? Left to right? A tree traversal is a specific order in which to trace the nodes of a tree. There are 3 common tree traversals. 1. in-order: left, root, right 2. pre-order: root, left, right 3. post-order: left, right, root. Department of Computer Science _ UHD 3
4 Binary Tree Traversal Methods Types of traversals Pre-order Visit root, traverse left child, traverse right child In-order Traverse left child, visit root, traverse right child The in-order traversal is probably the easiest to see, because it sorts the values from smallest to largest. Post-Order Traverse left child, traverse right child, visit root It is also called a depth-first search. Department of Computer Science _ UHD 4
5 Traversing Trees Pre-order traversal would give: A, B, D, E, C Tree A B C D E pre-order: root, left, right Department of Computer Science _ UHD 5
6 Traversing Trees In-order traversal would give: D, B, E, A, C in-order: left, root, right Department of Computer Science _ UHD 6
7 Traversing Trees Post-order traversal would give: D, E, B, C, A post-order: left, right, root Department of Computer Science _ UHD 7
8 Traversing Trees Level-order Traversal would give: A, B, C, D, E Department of Computer Science _ UHD 8
9 Traversing Trees Preorder: Root, then Children + A * B / C D Postorder: Children, then Root A B C D / * + Inorder: Left child, Root, Right child A + B * C / D + A * B / C D Department of Computer Science _ UHD 9
10 Preorder, Postorder and Inorder Pseudo Code Department of Computer Science _ UHD 10
11 Tree Traversal Example Ex. Write the 3 traversals of the given tree. In-order: Chewbacca, Han, Lando, Leia, Luke, Obi, Vader, Yoda Pre-order: Luke, Han, Chewbacca, Leia, Lando, Vader, Obi, Yoda Post-order: Chewbacca, Lando, Leia, Han, Obi, Yoda, Vader, Luke Department of Computer Science _ UHD 11
12 Illustrations for Traversals Assume: visiting a node is printing its data Preorder: Inorder: Postorder: Department of Computer Science _ UHD 12
13 Preorder Of Expression Tree / * e f a b c d / * + a b - c d + e f Gives prefix form of expression! Department of Computer Science _ UHD 13
14 Inorder Of Expression Tree Gives infix form of expression Department of Computer Science _ UHD 14
15 Postorder Of Expression Tree a b + c d - * e f + / Gives postfix form of expression! Department of Computer Science _ UHD 15
16 Some terminology of Binary Trees The successor nodes of a node are called its children The predecessor node of a node is called its parent The "beginning" node is called the root (has no parent) A node without children is called a leaf Department of Computer Science _ UHD 16
17 Some terminology of Binary Trees What is the max #nodes at some level i? The max # nodes at level i is where i = 0,1,2,...,L Department of Computer Science _ UHD 17
18 How to search a binary tree? (1) Start at the root (2) Search the tree level by level, until you find the element you are searching for or you reach a leaf. Department of Computer Science _ UHD 18
19 Binary Search Trees (BSTs) Binary Search Tree Property: The value stored at a node is greater than the value stored at its left child and less than the value stored at its right child Department of Computer Science _ UHD 19
20 Binary Search Trees (BSTs) Where is the smallest element? Ans: leftmost element Where is the largest element? Ans: rightmost element Department of Computer Science _ UHD 20
21 How to search a binary search tree? (1) Start at the root (2) Compare the value of the item you are searching for with the value stored at the root (3) If the values are equal, then item found; otherwise, if it is a leaf node, then not found Department of Computer Science _ UHD 21
22 How to search a binary search tree? (4) If it is less than the value stored at the root, then search the left subtree (5) If it is greater than the value stored at the root, then search the right subtree (6) Repeat steps 2-6 for the root of the subtree chosen in the previous step 4 or 5 Department of Computer Science _ UHD 22
23 Other Kinds of Binary Trees Full Binary Tree: A full binary tree is a binary tree where all the leaves are on the same level and every non-leaf has two children The first four full binary trees are: Department of Computer Science _ UHD 23
24 Examples of Non-Full Binary Trees These trees are NOT full binary trees: (do you know why?) Department of Computer Science _ UHD 24
25 Labeling of Full Binary Trees Label the nodes from 1 to n from the top to the bottom, left to right Department of Computer Science _ UHD 25
26 Thank you??? Department of Computer Science _ UHD 26
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Slides adapted from Professor Danny Heap course material winter17 CSC148-Section:L0301 Week#8-Friday Instructed by AbdulAziz Al-Helali a.alhelali@mail.utoronto.ca Office hours: Wednesday 11-1, BA2230.
Binary Trees. Height 1
Binary Trees Definitions A tree is a finite set of one or more nodes that shows parent-child relationship such that There is a special node called root Remaining nodes are portioned into subsets T1,T2,T3.